Section  is devoted to a detailed analysis
of invariant convex cones in the Lie algebra
\(\mathcal {V}({\mathbb {S}}^1)\)  of smooth vector fields on the circle and its
central extension, the Virasoro algebra \(\mathop {{\mathfrak {vir}}}\nolimits \) .
In particular we show that, up to sign, there are only two open
invariant cones in \(\mathcal {V}({\mathbb {S}}^1)\) . From this insight we derive that
the group \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  of orientation preserving diffeomorphisms
of \({\mathbb {S}}^1\)  has no non-trivial semibounded unitary representations,
which is derived from the triviality of all unitary highest weight
modules ([1]}). As one may expect from its importance
in mathematical physics, the situation is different
for the Virasoro group \(\mathop {{\rm Vir}}\nolimits \) .
For \(\mathop {{\rm Vir}}\nolimits \)  we prove a convexity theorem for
adjoint and coadjoint orbits which provides complete
information on invariant cones and permits us to
determine the momentum sets of the unitary highest weight representations.
In particular, we show that these, together with their duals, are precisely
the irreducible semibounded unitary representations.
Our determination of the momentum sets uses the complex analytic tools
from Section , which lead to a realization
in spaces of holomorphic
sections on the complex manifold \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+/{\mathbb {S}}^1\) .
This manifold has many interesting realizations.
In string theory it occurs as as a space
of complex structures on the based loop space
\(C^\infty _*({\mathbb {S}}^1,{\mathbb {R}})\)  ([2]}),
and Kirillov and Yuriev
realized it as a space of univalent holomorphic functions
on the open complex unit disc ([3]}, [4]}).
Its close relative \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)/\mathop {{\rm PSL}}\nolimits _2({\mathbb {R}})\)  can be identified
with the space of Lorentzian metrics on the one-sheeted hyperboloid
(cf. [5]}), which leads in particular to an interpretation
of the Schwarzian derivative in terms of a conformal factor.
